Nomvete cancelled out a superb opening goal from the Clever Boys’ Ryan Chapman to leave the score at 1-1 at the break.
He then notched up his second with just 12 minutes to play and Mpho Maleka added a third as the Birds continued their fine form – this is a fourth win from their last five matches.
But Wits will rue a penalty miss by Stanton Fredericks with the score at 2-1.
Wits got off to the best possible start, opening the scoring after just three minutes through Chapman, who rifled home a cracking left-footed shot from just outside the penalty area.
The Birds made it 1-1 in the 43rd minute when Nomvete bundled the ball home from close range, taking advantage of a mistimed header from the usually reliable Mbulelo Mabizela.
The teams continued to cancel each other out until the last 12 minutes when Swallows went ahead for the first time. Nomvete scored a fine solo goal, gliding beyond Derrick Spencer with his pace before beating Energy Murambadoro at his near post with a crisp finish.
The Clever Boys had an immediate opportunity to equalise when they were gifted a soft penalty. However, substitute Fredericks saw his casually-taken effort saved by Greg Etafia – the man who had been harshly penalised for a supposed foul on Sibusiso Vilakazi.
The Birds then killed off the game in the 81st minute, cutting Wits apart on the counter-attack beforeMaleka coolly finished off the chance to make it 3-1 and secure a fine away win for the Soweto side.